Similar Triangles/Polygons

Lesson Objective Lesson Success Criteria To learn about similarity in triangles, and other polygons Lesson Success Criteria Can identify and solve problems involving similar triangles Can work successfully with ratios in solving geometric problems Can solve problems involving polygon similarity

Similar Triangles – what are they? When two triangles are equiangular, then one triangle is an enlargement of the other – they are known as similar triangles. Their sides will be proportion, that is, the ratio of the lengths of the same sides is the same. 𝑋𝑌 𝐴𝐵 = 𝑌𝑍 𝐵𝐶

Similar Triangles - examples Here are some common examples of similar triangles. Note the parallel sides in the first two examples. Remember: Equiangular means equal angles.

Similar Triangles - calculation Identifying similar triangles is a skill, as you are not normally told this. You may need to use geometric reasons to prove similarity first. Identify the two equiangular triangles, if possible, draw them as two separate triangles Identify which sides are in the same relative position Apply appropriate ratios to help calculate unknown sides Be careful: Some figures may overlap – identify carefully the lengths required

Similar Triangles – problem 1 All angles are equiangular, therefore we have similar triangles. We are asked to calculate side length x. 𝑂𝑢𝑟 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 𝑋𝑌 𝐴𝐵 = 𝑌𝑍 𝐵𝐶 ∴ 32 20 = 𝑥 8 ∴ 𝑥= 32×8 20 =12.8𝑐𝑚

Similar Triangles – problem 2 Calculate the height of the tree. This is done using the shadow length, and a known height of another object. 𝑂𝑢𝑟 𝑟𝑎𝑡𝑖𝑜 𝑖𝑠 84 12 = ℎ 2 ∴ ℎ= 84×2 12 =14 𝑚

Similarity -polygons The same principles can be applied to any polygons that are similar: Corresponding angles are equal Corresponding sides are in proportion Following the same process as with triangles, you can through geometric reasoning solve for unknown sides. Remember: Corresponding means same position.

Practice From homework book Page 199 Ex F: Similarity